Academic Genealogy of Shoshichi Kobayashi and Individuals Who Inﬂuenced Him

Home , Burt Totaro, Carl Friedrich Gauss, Henri Cartan, Hisashi Kobayashi, Jakob Emanuel Handmann, Kenjiro Shoda

Progress in Mathematics, Vol. 308, 17–38 c 2015 Springer International Publishing

Hisashi Kobayashi

1. Introduction Professor Yoshiaki Maeda, a co-editor of this volume, kindly asked me to prepare an article on academic genealogy of my brother Shoshichi Kobayashi. In the olden times, it would have been a formidable task for any individual who is not in the same ﬁeld as the subject mathematician to undertake a genealogy search. Luck- ily, the “Mathematics Genealogy Project” (administered by the Department of Mathematics, North Dakota State University) [1] and Wikipedia [2] which document almost all notable mathematicians, provided the necessary information for me to write this article. Before I undertook this study, I knew very little about Shoshichi’s academic genealogy, except for his advisor Kentaro Yano at the Uni- versity of Tokyo and his Ph.D. thesis advisor Carl Allendoerfer at the University of Washington, Seattle. Shoshichi was well versed in the history of mathematics and wrote several tutorial books (in Japanese) (e.g., [3]) and an essay book [4]. In these writings, he emphasized the importance and necessity of studying the history of mathematics in order to really understand why and how some concepts and ideas in mathematics emerged. I am not certain whether Shoshichi was aware that one of his academic ancestors was Leonhard Euler (1707–1783) (see Table 1), whom he immensely admired as one of the three greatest mathematicians in human history. The other two of his choice were Carl Friedrich Gauss (1777–1855) and Archimedes (circa 287 BC–212BC). I was pleasantly surprised to ﬁnd that his academic ancestry includes not only Euler but also such a large list of celebrated mathematicians, as shown in Table 1: Gottfried Leibniz (1646–1716), Jacob Bernoulli (1655–1705), Johann Bernoulli (1667–1748), Jean d’Alembert (1717–1783), Joseph Lagrange (1736–1813), Pierre- Simon Laplace (1749–1827) and Simeon Poisson (1781–1840). In Section 2, I will summarize what I have found about some of the older ancestors of Shoshichi: Gottfried Leibniz, Nicolas Malebranche, Jacob Bernoulli, Johann Bernoulli and Leonhard Euler. It goes without saying that we would learn a great deal also by studying d’Alembert, Lagrange, Laplace and Poisson,allof 18 H. Kobayashi

Gottfried Wilhelm Leibniz (1666, Universität Leipzig) ? Nicolas Malebranche (1672, school unknown)

Jacob Bernoulli (1684, Universität Basel)

Johann Bernoulli (1694, Universität Basel)

Leonhard Euler (1726, Universität Basel) Jean Le Rond d'Alembert

Joseph Louis Lagrange Pierre-Simon Laplace

Simeon Denis Poisson (1800, École Polytechnique)

Michel Chasles (1814, École polytechnique)

Hubert Anson Newton (1885, Yale Univerity)

Eliakim Hastings Moore (1885, Yale University)

Oswald Veblen (1903, University of Chicago)

Tracy Yerkes Thomas (1923, Princeton University)

Carl Barnett Allendoerfer (1937, Princeton University)

Shoshichi Kobayashi (1956, University of Washington)

Table 1. Shoshichi Kobayashi’s Academic Ancestry

whom are household names, so to speak, to all scientists and engineers. But in the interest of space, I should ﬁnd another opportunity to discuss these mathematicians. In Section 3, I will make another important account of various individuals with whom Shoshichi interacted directly, although they do not appear in Shoshichi’s genealogical tree of Table 1. Academic Genealogy of Shoshichi Kobayashi 19

2. Some Great Mathematicians among Shoshichi’s Academic Ancestry As stated earlier, Table 1 provides an important segment of Shoshichi Kobayashi’s academic ancestry. At the top of Table 1 is Gottfried Wilhelm Leibniz (1646–1716), a prominent German mathematician and philosopher [5, 6]. Gottfried Leibniz developed the theory of differentiation and integration, independently of his contemporary, Isaac Newton (1642–1727) (see the Leibniz– Newton controversy on calculus [7]) and Leibniz’ no- tation for infinitesimal calculus has been widely used ever since it was introduced [4] (pp. 46–48). His Ph.D. thesis at age 20 (in 1666) was on “Disputa- tio arithmetica de complexionibus (Arithmetic dis- cussion of combinations)” under two advisors (not shown in Table 1) at the University of Leipzig: Jakob Thomasius (1622–1684), a philosopher, and Er- hard Weigel (1625–1699), a mathematician and as- tronomer. Thomasius’ advisor was Friedrich Leibniz (1597–1652), the father of Gottfried Leibniz, a professor of philosophy at Leipzig. Gottfried’s other advisor Weigel was an academic descendant of Nicolas Coper- nicus (1473–1543) (not shown in Table 1). Gottfried Wilhelm von During his stay in France around 1670, Leibniz Leibniz (1646–1716) met the Dutch physicist and mathematician Christian Source: http://www- Huygens (1629–1695). With him as mentor, Leibniz history.mcs.st-and.ac.uk/. made major contributions, including the discovery of Public domain. differential and integral calculus. According to the Mathematics Genealogy Project [6], Leibniz submitted another dissertation to Académie Royale des sciences de Paris in 1676 with Huygens as the advisor, although the type of degree is not given. Thus, we may claim Huygens as a mathematical ancestor of Shoshichi. Nicolas Malebranche (1638–1715) was a French Oratorian priest and ratio- nalist philosopher. The Mathematics Genealogy Project shows Malebranche as one of Leibniz’s two students [6], but neither his dissertation title nor his university is given. In Wikipedia [8] no mention is made regarding the advisor-student relationship between Malebranche and Leibniz, but their collaboration on physics and mathematics is briefly mentioned. Considering that Malebranche was eight years senior to Leibniz, it is quite possible that Malebranche was not an official student of Leibniz, but he learned mathematics and physics from Leibniz. We need more investigation to prove the genealogy link between the two. Malebranche introduced Guillaume de l’Hôpital (1661–1704) to Johann Bernoulli (see a paragraph below regarding an unusual contract and dispute between l’Hôpital and Johann Bernoulli). Jacob Bernoulli (1654–1705) [9] was the first mathematician in the famous Bernoulli family, of which Shoshichi gives a full account in Chapter “Mathe- matical families” (pp. 30–39) of [4] as well as in Section 4.1 (pp. 167–171) of 20 H. Kobayashi

“Chapter 4: Euler” of his book on Fermat and Euler [3]. Jacob made numerous contributions, and is well known, among other things, for the calculus of variations and the Bernoulli numbers. The Bernoulli numbers appear in various mathematical fields. Those who study probability theory (see, e.g., [10] which I recently published) learn the work of Ja- cob Bernoulli through Bernoulli’s Theorem (a.k.a. the weak law of large numbers), Bernoulli trials, Bernoulli distributions, etc. In 1657, the aforementioned Chris- tian Huygens published the first book on probability entitled De Ratiociniis in Ludo Aleae (On Reasoning in Games of Chance), a treatise on problems associated with gambling, motivated by Pascal and Fer- Jacob Bernoulli (1654– mat’s correspondence (see, e.g., [10]). Huygens’ book 1705) Source: Wikipedia. influenced Jacob Bernoulli who worked on probabil- Public domain. ity theory around 1684–1689. Much of Jacob’s work is found in his book Ars Conjectandi (The Art of Conjecture), published posthu- mously in 1713 by his nephew Nicholas Bernoulli (1687–1759). Johann Bernoulli (1667–1748) [11] who was 13 years younger than Jacob, studied mathematics from Jacob. Throughout Johann’s education at the Univer- sity of Basel, the Bernoulli brothers worked together on the newly discovered infinitesimal calculus. Shortly after the graduation of Johann, however, the two developed a jealous and competitive relationship [4, 11]. After Jacob’s death, Johann’s jealousy shifted to- wards his own talented son, Daniel Bernoulli (1700– 1782). Johann Bernoulli, introduced by Malebranche, was hired by l’Hôpital for tutoring in mathematics. In 1694, l’Hôpital made an unusual contract with Johann Bernoulli [4, 12] in exchange for an annual payment of 300 Francs that Johann would inform Johann Bernoulli l’Hôpital of his latest mathematical discoveries. Two (1667–1748) Source: years later l’Hôpital authored the first textbook on Wikipedia. Public infinitesimal calculus, Analyse des Infiniment Petits domain. pour l’Intelligence des Lignes Courbes (Analysis of the infinitely small to understand curves), including what is now known as l’Hôpital’s rule. After l’Hôpital’s death, Johann publicly revealed their agreement and claimed credit for portions of the text of Analyse. For a long time, however, his claims were not regarded as credible by many historians and mathematicians, because Johann had been previously involved in several disputes with his brother Jacob and his own son Daniel. In 1921, however, Paul Schafheitlin [13] discovered a manuscript of Johann’s lectures on differential calculus written during 1691–1692, substantiating Bernoulli’s account of the book’s origin. Academic Genealogy of Shoshichi Kobayashi 21

Leonhard Euler’s (1707–1783) father Paul Euler was a pastor, who studied in his youth mathematics from Jacob Bernoulli, and thus was a friend of the Bernoulli family. Leonhard studied mathematics from Johann Bernoulli and became a close friend of Johann’s sons Nicholas and Daniel. In 1725 Daniel Bernoulli was appointed the chairman of the mathematics/physics division of the St. Petersburg Academy of Sci- ences (today’s Russian Academy of Sciences), established that year by Pe- ter the Great who had consulted with Leibniz. Upon Daniel’s recommenda- tion, Euler was appointed to a position of physiology at the Academy in 1727 at age 20, and became a professor of physics in 1731. In 1733, Daniel Bernoulli returned to the University of Basel and Leonhard took over Daniel’s position as the head of the mathematics department at the Academy. In 1735 (age 28) he suffered a near-fatal Leonhard Euler (1707–1783). Source: Wikipedia. Portrait of Leonhard Euler fever and became almost blind in his painted by Jakob Emanuel Handmann right eye, but continued to be produc- (1718–1781). Public domain. tive in his research [3, 14]. Concerned about the political turmoil in Russia, triggered by the death of Empress Anna, Euler left Russia in 1741 to take up a position at Berlin Academy offered by Frederick the Great of Prussia. During the next 25 years’ stay in Berlin, Euler wrote over 380 articles, a book on functions (1748), and another on differential calculus (1755). Despite his enormous productivity and contributions to the Academy’s prestige, Euler was not entirely happy at Berlin [3, 14]: He did not get along well with King Frederick and his social circle of philosophers, including the likes of Voltaire (1694–1778). In 1766 when he was 59, Euler accepted an invitation to return to the St. Petersburg Academy (whose official name had been changed to the Imperial Acad- emy of Sciences since 1747), and spent the rest of his life in St. Petersburg. But his second stay in Russia was marred by a series of tragedies. By 1771 he had lost his left eyesight due to a cataract, he lost his home by a fire in St. Petersburg, and in 1773 he lost his wife Katharina (1707–1773) after 40 years of marriage. Despite these tragedies he remained amazingly productive. In 1775, he produced on aver- age one mathematical paper every week [15]. He suffered a cerebral hemorrhage on Sept. 18, 1783, and died at age 76. Euler is the only mathematician to have two numbers named after him: Eu- ler’s number e and Euler’s constant γ (a.k.a. Euler–Mascheroni constant), which 22 H. Kobayashi is defined as the limiting difference between the harmonic series and the natural logarithm [3, pp. 223–224]. Euler’s identity (which is a special case of Euler’s formula) eiπ + 1 = 0 is said to be the most beautiful mathematical theorem, because three basic arithmetic operations (addition, multiplication, and exponentiation) occur exactly once each and the identity also links five fundamental mathematical constants (0, 1, e, i, π). Euler’s contributions ranges from differential and integral (infinitesimal calculus), to number theory, graph theory, logic, astronomy, physics and philosophy. By calculating the zeta function at integers, Euler became the father of analytic number theory [3].

3. Individuals who Influenced Young Shoshichi In this section I will write about several individuals who directly influenced Shoshichi in his early career. Shoshichi wrote about Mr. Muneo Hayashi in his essays [17, 18, 19], who was his mathematics teacher and mentor in his 4th year (academic year 1947/48) at Nozawa Middle School in Nagano Prefecture. He writes [18] “After the school hours, Mr. Hayashi taught me various mathematics. Matrices and their equations felt like magic. . . . I accompanied Mr. Hayashi to a book store. We found books such as The- ory of Functions by Tanzo Takeuchi. I learned that Muneo Hayashi (1916– 2012). Source: Prof. there is a field of mathematics called ‘function the- Emer. Masaru Mitsuishi ory.’ Although it was the period when I felt hungry (Shinshu University, all the time, every day was exciting to me, thanks to Nagano, Japan). Mr. Hayashi who taught me new subjects of mathematics, one after another.” Soon after Shoshichi entered Ichikoh (the Number One Higher School) in Tokyo, a new educational system came in effect. After having attended Ichikoh for just one year, Shoshichi became eligible to take the entrance examination of Todai (the University of Tokyo) because one year that he skipped in Middle school was credited to his precollege schooling years. In his autobiographic essay [19] he fondly recalls the “Sugaku Kenkyukai” (Mathematics Research Group) he participated in during his freshman and sopho- more years, when he primarily learned advanced mathematics through “Rinkoh” (in which the group members gave lectures in turn). He writes: “At Komaba, I attended seriously language classes (English and German), but skipped mathematics classes, because I was busy studying in the Rinkoh. Even after I moved to the Hongo campus in my junior and senior years, I did not attend most mathematics classes. Attending a class would be valuable in reviewing what I already knew, but I could not absorb new materials by just attending the lecture. In order to understand new stuff, I had to pick an appropriate book myself and study it by spending enough hours. . . . ” The 17 text books that the math Academic Genealogy of Shoshichi Kobayashi 23 group studied at “Rinkoh” included Topologies (by Alexandroff–Hopf), Topolog- ical Groups (by Pontryagin), Survey of Algebra (in Japanese by Kenjiro Shoda), Survey of Linear Algebra (in Japan- ese, by Keizo Asano), Algebraic Num- ber Theory (in Japanese, by Teiji Tak- agi), and Fundamentals of Projective Geometry (in Japanese, by Hidetaka Terasaka). Professor Kentaro Yano (1912– 1993) [20, 21], Shoshichi’s advisor in his senior year, influenced his choice of differential geometry. Their close relationship continued even after Shoshichi left Japan. Sugaku Kenkyukai. With members of Yano graduated from the Univer- Sugaku Kenkyukai. Front from left: sity of Tokyo in 1934, and he studied Keijiro Yamazaki, Hideki Sakurai; Rear with Elie´ Cartan (1869–1951) in Sor- from left: Shuntaro Ito, Shoshichi bonne, Paris from 1936 to 38 on a two- Kobayashi. Source: Shoshichi Kobayashi, year scholarship. Shiing-Shen Chern From “Elementary School until Graduation from College” (in Japanese) (1911–2004) was also visiting Cartan, in “To Begin Studying Mathematics”, immediately after having earned his Vol. 1, (Nippon Hyoron Sha Co. Ltd.), Ph.D. from the University of Ham- pp. 136–151. burg. Chern became later instrumental in making U.C Berkeley a world center of geometry, where Shoshichi joined in 1962 upon invitation by Chern. Shoshichi describes his encounter with Yano in [19]: ”When I became a fourth year student at Todai, I had to choose a professor of “Seminar” (in which the professor and a small number of students study, discuss and give lectures in turn on some chosen topics). At the suggestion of Prof. Yukiyoshi Kawada, whom I had known since my Komaba campus days, I chose Prof. Yano’s seminar to study harmonic integrals.” Regarding his presentations at Yano’s seminar, Shoshichi recalls [19, 22]: “. . . The book on harmonic integrals by Hodge was difficult to understand. It was before André Weil’s 1952 paper on a simple proof of “de Rham’s theorem” by use of Cech cohomology. Needless to say, it was well before de Rham published the book Variétés Différentiables. Therefore, in Pro- Kentaro Yano fessor Yano’s seminar, I spoke several times concern- (1912–1993). Source: ing the 1946 paper by Bidal & de Rham [23] and other http://www-history. papers, by referring to the lecture note of de Rham & mcs.st-andrews.ac.uk. Kodaira given at Princeton’s Institute. . . . Public domain. 24 H. Kobayashi

“Professor Yano had studied at Princeton the method developed by Salomon Bochner (1899–1982), so in 1952, my senior year, he gave lectures on ‘Curvature and Betti numbers,’ based on his manuscript with Bochner under preparation. So we had an opportunity to learn the state-of-the art results on differential geometry. Furthermore, his lectures were related to my seminar topic, harmonic integrals. I could not have asked for more. His lectures also helped me get familiar with tensor analysis of, not only Riemannian manifolds, but also Kähler manifolds, and they were useful to my research in later years as well. Bochner’s method gave rise to a vanishing theorem for the cross section of a vector bundle, and Kodaira’s vanishing theorem is an outgrowth of Bochner’s method. I had several opportunities in later years to use Bochner’s method. It was 30 years later when I used this method to investigate the stability of a regular vector bundle. Professor Yano was working on transformation groups that preserve differential geometric structures (such as Riemann metric, affine connection, conformal connection), and lectured on these topics. Later, I started actively investigating transformation groups, which became one of my research topics for more than ten years.” Although Yano worked closely with Bochner while he was at Princeton, according to his biography [19], he was invited to Princeton’s Institute as an assistant to Oswald Veblen (1880–1960). Veblen taught at Princeton University from 1905 till 1932, when he was asked to help organize the Institute for Advanced Study, and became the first professor of the Institute and remained in that position until 1950 [24]. It is to be noted Veblen was the thesis advisor of Carl Allendoerfer, Shoshichi’s thesis advisor at the University of Washington in Seattle. Upon graduating from the University of Tokyo in 1953. Shoshichi studied for one year in France, first at the University of Paris and then at the University of Strasbourg on the French Government’s Scholarship. He attended a seminar by Henri Cartan (1904–2008) on several complex variables in 1953/1954 [25]. Henri Cartan was a son of Elie´ Cartan who was the thesis advisor of Kentaro Yano seventeen years earlier. Dur- ing his stay in Paris, Shoshichi also listened to seminar talks by Karl Stein (1913–2000) and AndréLich- nerowicz (1915-1998) at Collège de France. He also interacted with Marcel Berger (1927–), Paulette Liber- mann (1919–2007), Warren Ambrose (1914–1995) who was on a sabbatical leave from MIT, and Katsu- mi Nomizu, a Japanese mathematician, eight years senior to Shoshichi. Katsumi Nomizu (1924–2008) [27] graduated Henri Cartan (1904– 2008). Source: from Osaka University with a master of science degree Oberwolfach Photo in 1947, and did his postgraduate study at Sorbonne, Collection. c Gerd and continued his study under Shiing-Shen Chern at Fischer, München, 1985. the University of Chicago and received his Ph.D. in Reprinted with perm. 1953. He was visiting France as a post-doctoral fellow Academic Genealogy of Shoshichi Kobayashi 25 at the Centre national de la recherche scientifique (CNRS) from 1953 to 1954. Nomizu wrote in March 2007, less than two years before his death, for the occasion when he and Shoshichi received the publication award from the Mathematical Society of Japan [28]: “. . . While we (Shoshichi and Nomizu) enjoyed all this (i.e., the lectures by Henri Cartan, André Lichnerowicz et al.) in Paris, we felt we should work with concentration in a small nicer environment. We chose Strasbourg where Charles Ehresmann (1905–1979) and Jean-Louis Koszul (1921–) were teaching. We wanted to reorganize some of the classical results, for example, in the 1917 paper by Levi- Civita and Ricci, Elie´ Cartan’s results on symmetric spaces (Riemannian or affine), classical surface theory and general submanifold theory, relations with trans- formations, etc. . . . I want to say how Kobayashi- Nomizu was made into a book. Professor Lipman Bers (1914–1993), Editorial Board of Wiley Interscience Publishers, asked Professor Yasuo Akizuki for a suggestion on potential authors and my name came up. Bers approached me, I immediately thought “alright” Katsumi Nomizu if Kobayashi can help as coauthor. He said yes. We (1924–2008). Source: were 3,000 miles apart (note: Nomizu was with Brown Oberwolfach Photo University after 1960). . . . Those were the days when Collection. c Dirk Ferus, we had an electric typewriter at best. Volume I ap- Berlin, 1982. Reprinted peared in 1963, and Volume II in 1969.” with permission. Shoshichi also wrote for the same occasion [29]: “In the olden days, we had to begin with an explanation of ‘curvature’ when we talked on differential geometry in a colloquium. ‘Connection’ was known to only those who specialized in differential geometry. But since around the end of the 1970s when gauge theory blossomed, those who studied topology and mathematical physics also became interested in the theory of connection of fiber bundles. Thanks to that trend, our book which emphasized the connection theory for fiber bundles by bringing that topic to the beginning Katsumi Nomizu and Shoshichi of the volume came to be frequently re- Kobayashi. Source: Shoshichi Kobayashi, “My Teachers, My Friends and My ferred to. When I was a student, there Mathematics: The Period when I studied were few books on mathematics, and in America” (in Japanese), Sugaku everyone was reading the same books. Seminar (Nippon Hyoron Sha Co. Ltd.), Such books had a long life, and we July 1982 21(7), pp. 55–58. (Republished wanted to write books that would last in February 2013: Special Issue: Shoshichi for twenty to thirty years. Luckily our Kobayashi). 26 H. Kobayashi book was selected into Wiley Classic Library Series and has survived until today, and has the honor of receiving this Award. Now I have mixed feelings: While on one hand I wish to live longer than this book by paying attention to my own health, I wish on the other hand that this book won’t go out of print for another twenty to thirty years.” While they were together at Strasbourg in 1954, Nomizu stirred up Shoshichi, saying “Why don’t you study in the U.S., instead of returning directly to Japan?” Since Shoshichi felt that he needed about a year to make a doctoral thesis out of the results he obtained at Strasbourg, he became seriously interested in following Nomizu’s suggestion. He wrote to the University of Seattle in Washington, where Carl Allendoerfer (who gave the first proof of the Gauss–Bonnet theorem) was, and to the University of Chicago, where S.S. Chern (who gave a simpler proof of the same theorem) was. No sooner had the application form from a secretary of the Department Chair’s office of Chicago arrived than Shoshichi received a letter from Allendoerfer (the then Department Chair) offering him an assistantship. So he jumped at this opportunity, without thinking of anything else [26]. Shoshichi left France in early September 1954 by the “Ile de France” Ocean Liner, and arrived in New York five days later. He writes in [26]: “As our ship approached the Statue of Liberty, the skyline of Manhattan, numerous automobiles on the highway along Hudson River came into sight. New York looked like a huge and advanced city, compared with Tokyo and Paris. . . . With 50 dollars in my pocket, I could not afford sightseeing of New York, so after spending one night in New York, I took a Greyhound Bus straight to Seattle, where I arrived three days later.” Shoshichi describes his first meeting with Carl Allendoerfer (1911–2004) [26]: “. . . Having been the department chair for a while, he was a pragmatic man. When I went to greet him upon my arrival, he surprised me by starting our conversation by asking ‘Do you have enough money for now? If not, I could pay your assistantship in advance.’ In truth I was short of money but felt that it would be rather disgrace- ful to borrow money immediately after the arrival, soIendedupwithsaying‘IamOK.’Afterwewere talking for a while, he probably found that with my poor English I could not serve well as a TA (teaching assistant), so he changed my title from TA to RA Carl B. Allendoerfer (researchassistantship)immediately.... TheMath- (1911–1974). Source: ematics Department of the University of Washington Wikipedia. Public improved significantly during his 20 year tenure as the domain. chair....” Allendoerfer [30] graduated from Haverford College of Pennsylvania in 1932, and went to Oxford as a Rhodes scholar from 1932 to 1934. He received his Ph.D. Academic Genealogy of Shoshichi Kobayashi 27 from Princeton in 1937, with Oswald Veblen as the advisor. He became well known for his 1943 paper with André Weil on a proof of the Gauss–Bonnet theorem [31]. He was with the Institute for Advanced Study, Princeton in 1948–49, and in 1951 he became professor and soon af- terthechairoftheMathematicsDe- partment at the University of Wash- ington. Shoshichi writes about his Seat- tle days [26] “Since I thought that ifIwriteupwhatIworkedonin France, that will make a Ph.D. thesis, Carl Allendoerfer with Kentaro Yano in 1961. Source: Shoshichi Kobayashi, “My I felt at ease after having passed the Teachers, My Friends and My language exams (German and French) Mathematics: The Period when I studied and the Ph.D. qualifying exam (which in America” (in Japanese), Sugaku tested mathematics at the level of what Seminar (Nippon Hyoron Sha Co. Ltd.), we learned in the third and fourth July 1982 21(7), pp. 55–58. (Republished years at undergraduate and in the first- in February 2013: Special Issue: Shoshichi year graduate). So I spent my time by Kobayashi). thinking about holonomy and transformation groups that I learned from Dr. Nomizu and Lichnerowicz while I was in France. I also studied complex manifolds and attended Italian language classes. There were not many interesting seminars, so I got a little bit bored. In 1955, my second year, Hsien Chung Wang (Ph.D. University of Manchester, 1948) [32] came from Princeton (1954–55). He became well known about two years earlier for his work on classification of simply connected compact complex homogeneous spaces. He was a very versatile mathematician working in such a wide range of fields as topology, differential geometry, and Lie groups. His lecture on complex manifolds was perspicuous . . . ” Shoshichi completed his Ph.D. thesis in the spring of 1956, in less than two years after his arrival at Seattle. The dissertation title was Theory of Connections. He was then appointed a post-doctoral fellow at the Institute for Advanced Study, Princeton (1956–58). Albert Einstein (1879–1955) died a year before Shoshichi moved to Princeton. Shoshichi got married to Yukiko Ashizawa in May 1957 after his Wedding of Shoshichi first year at Princeton and Yukiko had completed her with Yukiko Ashizawa, study at Seattle. Their first daughter Sumire was born St. Mark Cathedral, in 1958 at Princeton. Sumire studied biochemistry at Seattle, May 11, 1957. Princeton University, where she met her future hus- Source: Kobayashi Family band Philip Chou, who was an electrical engineering Photo Album. 28 H. Kobayashi student. They now live outside Seattle where Phil is a Manager at Microsoft Re- search. Their first son Andrew got his BS and MS degrees in Computer Science from Stanford in 2013 and now works for Coursera, a major player in MOOC (Massive Open Online Course). Their second son Brendan is currently a junior in Princeton’s Dept. of Mechanical & Aerospace Engineering. Sumire’s younger sister Mei studied chemistry at Princeton. So there are a lot of Princetonians in the Kobayashi family, myself included. In the fall of 1958 Shoshichi moved to MIT as a Research Associate. Warren Ambrose (1914–1995), whom Shoshichi had met in Paris five years earlier, and his associate Isadore M. Singer (1924–) were Shoshichi’s sponsors at MIT [33]. In the fall of 1959, he received an offer of Assistant Professorship from UC Berkeley, but because he had entered the U.S. with a J-1 visa (exchange visa) when he moved from Strasbourg to Seattle, the U.S. immigration law required him to leave the U.S. for at least two years. With help from Berkeley, he found an assistant professor position at the University of British Columbia in Vancouver, Canada, where he taught for two and half years from January 1960 until the summer of 1962. In September 1962 he finally joined the faculty of Berkeley as Assistant Professor, was promoted to Associate Professor a year later, and then to Full Pro- fessor in 1966. He was attracted to Berkeley, because one of the greatest differential geometers of the 20th Century Shiing-Shen Chern (1911–2004) moved from the University of Chicago to Berkeley in 1960 with a mandate to build a geometry group at Berkeley. Chern was born in China, entered Nankai Uni- versity in 1926 at age 15 and received his B.Sc. in 1930, then went to teach at Tsinghua University in Beijing, while he enrolled there as a candidate for master’s degree, which he obtained in 1934. That was Chern, Shiing-Shen the first master’s degree in mathematics ever issued (1911–2004). Source: in China [34]. In 1934 he went to the University of Oberwolfach Photo Hamburg, Germany on a scholarship and received his Collection. c Gerd Ph.D. in 1936. His thesis advisor Wilhelm Blaschke Fischer, München, 1985. (1885–1962) recommended Chern to study with Elie´ Reprinted with perm. Cartan in Paris, where he met Kentaro Yano from Japan, who later became Shoshichi’s advisor at Tokyo. After returning to China, Chern taught in Tsinghua again from 1937 till 1943, and then he was invited by Princeton’s Institute (1943–46). Chern became well known for providing a simpler proof of the Gauss–Bonnet Theorem in 1944 [35] than Allendoerfer–Weil’s proof published a year earlier [31]. He introduced what is called the Chern classes, which are characteristic classes associated with complex vector bundles. Shoshichi’s expository article [33] and his memoir of Chern [36] provide a rather comprehensive account of Chern’s profound contributions to modern differential geometry. Academic Genealogy of Shoshichi Kobayashi 29

Chern taught at the University of Chicago from 1949 till 1960. Katsumi No- mizu was his first student, and Shoshichi’s longtime colleague at Berkeley Joseph Wolf [37] received his Ph.D. under Chern in 1959. Chern produced 31 Ph. D students at UC Berkeley, including Alan Weinstein in 1967 [38], another colleague of Shoshichi at Berkeley. Shing-Tung Yau [39], a 1982 Fields medalist, was also Chern’s student (1971). Shoshichi wrote two expository articles on Yau’s contributions in Japanese mathematical journals [40, 41]. 4. Shoshichi’s Students and Collaborators Shoshichi produced 35 Ph.D. students, all at Berkeley. Table 2 shows his students in the chronological order of their years of Ph.D. degrees granted. Added also are their current affiliations, wherever known. The numbers of their descendants in the right-most column are based on the data obtained from the Genealogy Project. These numbers will obviously change in the future, so the numbers given here serve as lower bounds. As of this writing, Shoshichi has 71 descendants. The fact that Shoshichi produced as many as 35 Ph.D.’s during his appointment of 34 years (from 1960 to 1994) at Berkeley is a testimony of his commitment to education. Shown on the next page is a photo taken in the fall of 1992, when several of Shoshichi’s students and some of the participants in the MSRI (Mathematical Sciences Research Institute) Program on Algebraic Geometry (organized by Shige- fumi Mori of Kyoto University, 1990 Fields medalist) got together and celebrated Shoshichi’s 60th birthday at Codornices Park in Berkeley.1 A special volume entitled Geometry and Analysis on Complex Manifolds: Festschrift for Professor S. Kobayashi’s 60th Birthday [43] (eds. Toshiki Mabuchi, Junjiro Noguchi and Takushiro Ochiai) was published in 1994, to which several of Shoshichi’s former students and other mathematicians contributed their original work, like this memorial volume. The contributors were: Marco Abate and Giorgio Patrizio, Shigetoshi Bando & Yum-Tong Siu, Ichiro Enoki, Zhuang-dan (Daniel) Guan, Yoichi Imayoshi, Toshiki Mabuchi, Hong-Jong Kim, Ying-chen Li, Junjiro Noguchi, Takeo Ohsawa, Yongbin Ruan, Satoru Shimizu,andBurt Totaro. Shoshichi’s publication list is posted on the Shoshichi Kobayashi Memorial website [44] including his technical books, tutorial articles and essays published in journals, and an essay book. A list of his 150 plus publications, sorted in chronological order, can be found at [45]. By reviewing this list, we can find his research collaborators. Takushiro Ochiai (Ph.D. University of Notre Dame 1969; Advisor, Tadashi Nagano) has 14 joint papers with Shoshichi: (from 1970 to 1982). Tadashi Nagano (Ph.D. University of Tokyo 1959; Advisor Kentaro Yano) has 10 joint papers with Shoshichi. Nagano and Shoshichi were together in Prof. Yano’s Seminar, thus they are academic siblings, so to speak, and Ochiai is an academic nephew of Shoshichi. Nagano gives a comprehensive survey (11 pages) of Shoshichi’s work in [46], which he prepared for the occasion when Shoshichi received the Geometry Prize of the Mathematical Society of Japan in 1987.

1For more photos of the party, click on http://mathcs.holycross.edu/∼ahwang/misc/kobayashi 60th birthday.zip 30 H. Kobayashi

At Codornices Park in Berkeley, fall 1992. Reprinted by courtesy of Andrew D. Hwang (College of the Holy Cross). c Andrew D. Hwang. Reprinted with permission. First row: 1. H.C. Yang, 2. Yungbin Ruan, 3. Yukiko Kobayashi, 4. Shoshichi, 5. Nina Morishige, 6. Keiji Oguiso (Osaka University). Second row: 7. Andrew D. Huang (1993), 8. Shigefumi Mori (Kyoto University, 1990 Fields medalist), 9. Liaw Huang (1993), 10 Ying-Chen Li (1991), 11. Wang Sang Koon, 12. Zhuang-dan (Daniel) Guan (1993). Third row: 13. Keizo Hasegawa (1987), 14. Masanori Kobayashi (Tokyo Metropolitan University), 15. Hajime Tsuji, 16. Raphael Laufer (1997).

Other joint authors include Geneviève Avérous, W.M. Boothby, M.P. do Carmo, S.-S. Chern, Samuel I. Goldberg, Hsin Chu, Jun-ichi Hano, Camilla Horst, Masahisa Inoue, Peter Kiernan, Katsumi Nomizu, Yoshihiro Ohnita, Takeshi Sasaki, Eriko Shinozaki, Masaru Takeuchi, H.C. Wang,andHung-Hsi Wu. Shoshichi’s 10th Ph.D. student, Michael Minovitch (1970) [47] is known as a “planetary pioneer” for his gravity-assisted trajectory theory that was used by NASA in designing energy efficient trajectories of such interplanetary spacecrafts as Mariner 10 in its voyage to Venus and Mercury in 1973 and Voyager in its Plan- etary Grand Tour in 1976. His Ph.D. thesis under Shoshichi was entitled Mathe- matical Methods for the Design of Gravity Thrust Space Trajectories. “Credit for the first rigorous analysis of the concept after the start of the Space Age belongs to Michael Minovitch,” writes David Portree, a spaceflight historian [48]. Minovitch’s 1961 Jet Propulsion Laboratory internal document, “A method for determining Academic Genealogy of Shoshichi Kobayashi 31 interplanetary free-fall reconnaissance trajectories,” provided the first numerical solution to the three-body problem, one of the famous unsolved mathematical problems. His numerical solution offered the theoretical possibility of free and un- limited space travel anywhere in the entire solar system. He was interviewed in BBC’s Science Program in October 2012, which can be viewed on You Tube [49]. Shoshichi’s 25th Ph.D. student Burt Totaro (1989) [50] recently moved from the University of Cambridge, UK to UCLA. At Cambridge he was the Lowndean Professor of Astronomy and Geometry. “Burt is best known for his 1997 paper, ‘The Chow ring of a classifying space.’ The most impressive aspect of Burt’s work is its remarkable breadth and depth, encompassing algebraic geometry, represen- tation theory, topology, and combinatorics” says Shoshichi’s academic brother, James Carrell (Ph.D. 1967, University of Washington; Advisor, Carl Allendoer- fer), who is a professor at the University of British Columbia.

5. Shoshichi Kobayashi’s Friends Takushiro Ochiai’s article in this volume “In Memory of Professor Shoshichi Koba- yashi” gives a comprehensive account of Shoshichi’s contributions as a researcher and a teacher. The February 2013 issue of Japanese popular mathematical jour- nal, Sugaku Seminar, (Nippon-Hyoron-Sha) was the special issue dedicated to Shoshichi Kobayashi. The contributors were Takushiro Ochiai, Hisashi Kobayashi, Masao Hattori, Ichiro Satake, Keizo Hasegawa, Yoshiaki Maeda, Junjiro Noguchi, Makiko Tanaka,andToshiki Mabuchi. Shoshichi’s July 1982 essay [26] was repub- lishedinthisissue. The Shoshichi Kobayashi Memorial Website [51] provides an archival record of Shoshichi’s biography, books and articles, photo gallery, and the remembrance speeches delivered at his Memorial Services held at the chapel of Sunset View Cemetery, El Cerrito, California on September 8, 2012 (Hisashi Kobayashi, Mei Kobayashi, Alan D. Weinstein, Arthur E. Ogus) [52]. The webpage also posts a condolence message from Heisuke Hironaka (1970 Fields medalist) & Wakako Hironaka [53]. On May 22–25, 2013, Geometry and Analysis of Manifolds: A Memorial Sym- posium for Professor Shoshichi Kobayashi was held at the Mathematical Science Building of the University of Tokyo (Komaba campus), organized by Takushiro Ochiai, Yoshiaki Maeda and others, and hosted by Dean Takashi Tsuboi.Thesym- posium was attended by more than 130 mathematicians [54]. The invited speakers were: Ichiro Enoki, Akito Futaki, Gary Jensen, Shinichiro Matsuo, Reiko Miyaoka, Jo¨el Merker, Hiraku Nakajima, Junjiro Noguchi, Takeo Ohsawa, Makiko Tanaka, Hajime Tsuji, Paul Vojta and Shing-Tung Yau. Programs and abstracts are found in [55] On May 25, 2013, following the above symposium, a “Memorial Reception for Prof. Shoshichi Kobayashi,” also organized by Takushiro Ochiai and Yoshiaki Maeda, was held also at the Komaba campus, with more than one hundred participants [56]. The speeches of the following individuals are also posted (in both Eng- lish and Japanese) at the above website: Takushiro Ochiai, Takashi Tsuboi, Paul 32 H. Kobayashi

Table 2a. Academic Descendants of Shoshichi Kobayashi, part 1 Name Year of Ph.D. Current Affiliation Nr. of Descendants 1. Sebastian Su Koh 1964 West Chester University (Retired) 2. Francis Joseph Flaherty 1965 Oregon State University (Emeritus) 5 3. Vivian Yoh Kraines 1965 Meredith College 4. Gary Richard Jensen 1968 Washington Univ. St. Louis (Emeritus) 10 5. Myung He-Son Kwack 1968 Howard University (Emeritus) 1 6. John Robert Zumbrunn 1968 Equinox Fund Management LLC 7. Carlos Edgard Harle 1969 University of São Paulo, Brazil (Emeritus) 8. Peter Kiernan 1969 University of British Columbia (Emeritus) 9. John Douglas Moore 1969 University of California, Santa Barbara 3 10. Michael Andrew Minovitch 1970 Phaser Telepulsion Inc., Los Angeles 11. Roy Hiroshi Ogawa 1971 University of Nevada, Las Vegas (Retired) 12. Geneviève Maynadier Avérous 1974 CNAM (Conservatoire national des arts et métiers) 13. Rune Zelow Lundquist 1977 In Oslo, Norway 14. Toshiki Mabuchi 1977 Osaka University, Japan 15. Michael Jay Markowitz 1979 Information Security Corporation 16. Andrew Joseph Balas 1980 University of Wisconsin-Eau Claire (Deceased in 2003) 17. Blaise Grayson Morton 1981 University of Minnesota & Whitebox Advisors LLC 18. Jay Alan Wood 1982 Western Michigan University Academic Genealogy of Shoshichi Kobayashi 33

Table 2b. Academic Descendants of Shoshichi Kobayashi, part 2 Name Year of Ph.D. Current Affiliation Nr. of Descendants 19. Jae-Hyun Yang 1984 Inha University, Korea 8 20. Ahmad Zandi-Bami 1984 Ulysses Strategic Services 21. Hong-Jong Kim 1985 Seoul National University 22. Ricardo Francisco Vila-Freyer 1986 Centro de Investigación en Matemáticas, Mexico 23. Keizo Hasegawa 1987 Niigata University, Japan 24. Alexander John Smith 1987 University of Wisconsin, Eau Claire 25. William Charles Jagy 1988 University of Texas at Austin (formerly) 26. Burt James Totaro 1989 University of California at Los Angeles 9 27. Ivona Maria Grzegorczyk 1990 California State University, Channel Islands 28. Janis Marie Oldham 1990 Agricultural and Technical College of North Carolina 29. Ying-Chen Li 1991 JP Morgan Chase 30. David Warren Gomprecht 1993 The Dalton School, New York City 31. Zhuang-dan (Daniel) Guan 1993 University of California, Riverside 32. Liaw Huang 1993 Terry Consulting 33. Andrew David Hwang 1993 College of the Holy Cross 34. Dae Yeon Won 1995 Duksung Women’s University, Korea 35. Raphael Laufer 1997 Systems Planning and Analysis, Inc. 34 H. Kobayashi

Vo jta, Nobuo Naito, Yutaka Katase, Shigetoshi Kuroda, Gary Jensen, Shing-Tung Yau, Shuji Hosoki, Eriko Shinozaki, Toshiki Mabuchi, Akiko Ashizawa, Hisashi Kobayashi and Yukiko Kobayashi. Gary Jensen (Ph.D. Berkeley 1968), Shoshichi’s fourth student and a professor emeritus at Washington University, St. Louis, gave a very touching story about his relation with Shoshichi [57]. Shing-Tung Yau (Ph.D. Berkeley 1971; Advisor, S.-S. Chern) also gave a very moving speech [58] and com- posed a Chinese poem in honor of Shoshichi [59]. A memoir by Yoshihiko Suyama (in Japanese only), and a memoir by Noboru Naito (Shoshichi’s classmate at Nozawa Middle School) are also found in [53]. The December 2013 issue of ICCM Notices [60] devotes 21 pages to “In Mem- ory of Shoshichi Kobayashi,” contributed by Shing-Tung Yau, Hisashi Kobayashi, Noboru Naito, Takushiro Ochiai, Hung-Hsi Wu, Blaine Lawson and Joseph A. Wolf. The December 2014 issue of Notices of AMS [61] will carry “Remembering Shoshichi Kobayashi,” (Gary R. Jensen, coordinating editor). The contributors are Toshiki Mabuchi, Takushiro Ochiai, Joseph A. Wolf, Hung-Hsi Wu, Robert Greene, Gary R. Jensen, Myung H. Kwack, Eriko Shinozaki, Hisashi Kobayashi, and Mei & Yukiko Kobayashi.

Acknowledgment A number of individuals helped me locate Shoshichi’s 35 students. My special thanks go to Professors James Carrell, Keizo Hasegawa, Andrew Hwang, Gary Jensen, Ivona Grzegorczyk, Zhuang-dan Guan, Toshiki Mabuchi, Shigefumi Mori and Arthur Ogus. The author also thanks Brian L. Mark, Phil Chou and Sebastian Koh for their careful reading of the manuscript and editorial suggestions. Finally, this work would have been impossible, were it not for the “Mathematics Genealogy Project” and Wikipedia.

References [1] Mathematics Genealogy Project: http://genealogy.math.ndsu.nodak.edu/ [2] Wikipedia: http://en.wikipedia.org/wiki/Wikipedia [3] Shoshichi Kobayashi, Understanding Euler and Fermat (in Japanese), Kodansha Publishing Co., 2003. [4] Shoshichi Kobayashi, Mathematicians Who Lost Their Faces: Essays in idleness on mathematics (in Japanese), Iwanami Publisher, July 2013. [5] Gottfried Wilhelm Leibniz, Wikipedia: http://en.wikipedia.org/wiki/Gottfried Wilhelm Leibniz [6] Mathematics Genealogy Project: Gottfried Wilhelm Leibniz http://www.genealogy.ams.org/id.php?id=60985 [7] Leibniz-Newton Calculus Controversy: http://en.wikipedia.org/wiki/Leibniz%E2%80%93Newton calculus controversy [8] Nicolas Malebranche, Wikipedia: http://en.wikipedia.org/wiki/Nicolas Malebranche [9] Jacob Bernoulli, Wikipedia: http://en.wikipedia.org/wiki/Jacob Bernoulli Academic Genealogy of Shoshichi Kobayashi 35

[10] Hisashi Kobayashi, Brian L. Mark and Willam Turin, Probability, Random Processes and Statistical Analysis, Cambridge University Press, 2012. [11] Johann Bernoulli, Wikipedia: http://en.wikipedia.org/wiki/Johann Bernoulli [12] Guillaume de l’Hôpital, Wikipedia: http://en.wikipedia.org/wiki/Guillaume de l’H%C3%B4pital [13] Paul Schafheitlin, “Vorwort zur Vorlesunguber ¨ das Rechnen mit Differentialen von Johann Bernoulli (1691/92)”: https://www.uni-due.de/imperia/md/content/didmath/ag jahnke/bern vorw.pdf [14] Leonhard Euler, Wikipedia: http://en.wikipedia.org/wiki/Leonhard Euler [15] B.F. Finkel, “Biography – Leonhard Euler,” The American Mathematical Monthly, Vol. 4, No. 12, pp. 297–302. December 1897. http://www.jstor.org/stable/2968971 [16] Shoshichi Kobayashi, “Mathematics,” in Thirst for Beauty Vol. 2, Kadokawa-Bunko. pp. 107–131, May 2012. [17] Shoshichi Kobayashi, “Mr. Muneo Hayashi: My Mathematics Teacher in Middle School” (in Japanese), in The Mathematician I Luckily Encountered, Sugaku Shobo, June 2011, pp. 61–67. [18] Shoshichi Kobayashi, “Deeply Impressed by a Beautiful Theorem,” in Sugaku Sem- inar (Nippon Hyoron Sha Co. Ltd.), May 1973. [19] Shoshichi Kobayashi, “From Elementary School till Graduation from College” (in Japanese), in To Begin Studying Mathematics, Vol. 1, (Nippon Hyoron Sha Co. Ltd.), January 2006 (1st edition) pp. 136–151. [20] Kentaro Yano: Biography: http://www-history.mcs.st-and.ac.uk/Biographies/Yano.html [21] Kentaro Yano, Wikipedia: http://en.wikipedia.org/wiki/Kentaro Yano (mathematician) [22] Shoshichi Kobayashi, “When I was in the Fourth Year at University” (in Japanese), Sugaku Seminar (Nippon Hyoron Sha Co. Ltd.), Special Issue: My Encounter with Mathematics, April 1987, p. 51. [23] Pierre Bida and Georges de Rham, Les forms différentielles harmoniques, Comment. Math Helv. 19 (1946), 1–49. [24] Oswald Veblen, Wikipedia: http://en.wikipedia.org/wiki/Oswald Veblen [25] Shoshichi Kobayashi, “My Memory of Professor Henri Cartan,” Notices of the AMS, Volume 57, No. 8, pp. 954–955. [26] Shoshichi Kobayashi, “My Teachers, My Friends and My Mathematics: The Period when I studied in America” (in Japanese), Sugaku Seminar (Nippon Hyoron Sha Co. Ltd.), July 1982 21(7), pp. 55–58. (Republished in February 2013: Special Issue: Shoshichi Kobayashi). [27] Katsumi Nomizu, Wikipedia: http://en.wikipedia.org/wiki/Katsumi Nomizu [28] Katsumi Nomizu, “Message as a recipient of the 2007 Publication Award from the Mathematical Society of Japan,” March 2007. http://mathsoc.jp/publication/tushin/1202/nomizu.pdf, [29] Shoshichi Kobayashi, “Message as a recipient of the 2007 Publication Award from the Mathematical Society of Japan (in Japanese),” March 2007. http://mathsoc.jp/publication/tushin/1202/kobayashi.pdf 36 H. Kobayashi

[30] Carl Barnett Allendoerfer, Wikipedia: http://en.wikipedia.org/wiki/Carl B. Allendoerfer [31] Carl B. Allendoerfer and Andr´e Weil, “The Gauss–Bonnet Theorem for Riemannian Polyhedra,” Transactions of the American Mathematical Society, Vol. 53, no. 1, (January 1943), pp. 101–129. [32] Hsien Chung Wang, Biography http://www-history.mcs.st-and.ac.uk/Biographies/Wang.html [33] Shoshichi Kobayashi “Trends in Modern Geometry (1): Chern” (in Japanese), Sug- aku Seminar (Nippon Hyoron Sha Co. Ltd.), April 2003, 42(4), pp. 66–71. [34] Shiing-Shen Chern, Wikipedia: http://en.wikipedia.org/wiki/Shiing-Shen Chern [35] Shiing-Shen Chern, “A simple intrinsic proof of the Gauss–Bonnet formula for closed Riemannian manifolds,” Ann. of Math. (2) 45 (1944), 747–752. [36] Shoshichi Kobayashi, “Remembrance of Professor Shiing-Shen Chern” (in Japanese) in Mathematical Communications, (Japan Mathematics Society), November 2005, 10(3), pp. 21–29. http://mathsoc.jp/publication/tushin/1003/kobayashi.pdf [37] Joseph Wolf, Wikipedia, http://en.wikipedia.org/wiki/Joseph A. Wolf [38] Alan Weinstein, Wikipedia: http://en.wikipedia.org/wiki/Alan Weinstein [39] Shiing-Tung Yau, Wikipedia: http://en.wikipedia.org/wiki/Shing-Tung Yau [40] Shoshichi Kobayashi, “Fields Prize Story: Shing-Tung Yau” (in Japanese), Sugaku Seminar (NipponHyoron Sha Co. Ltd.), March 1983, 22(3), pp. 16–19. [41] Shoshichi Kobayashi, “Achievements of Mr. Shing-Tung Yau” (in Japanese), Sugaku (Mathematical Society of Japan), February 1983, 35(2), pp. 121–127. [42] Shoshichi Kobayashi, Mathematics Genealogy Project: http://genealogy.math.ndsu.nodak.edu/id.php?id=22141 [43] T. Mabuchi, J. Noguchi and Ochiai (eds.), Geometry and Analysis on Complex Manifolds: Festschrift for Professor S. Kobayashi’s 60th Birthday, World Scientiﬁc Publishing Co., River Edge, NJ 1994, ISBN: 981-02-2067-7. [44] Shoshichi Kobayashi publication website: http://www.shoshichikobayashi.com/books-articles-essays/ [45] Shoshichi Kobayashi publications (in chronological order): http://www.shoshichikobayashi.com/wp-content/uploads/2013/04/ ShoshichiKobayashi Publications and RelatedWorks.pdf [46] Tadashi Nagano, “On recent developments of geometry: Prof. Shoshichi Kobayashi and his work,” Sugaku, 41(1): pp. 64–75, 1989. [47] Michael Minovitch, Wikipedia http://en.wikipedia.org/wiki/Michael Minovitch His website: http://www.gravityassist.com/ [48] David S.F. Portree, “Castles in space: a 50-year survey of gravity-assist space travel.” http://www.wired.com/wiredscience/2012/12/castles-in-space-1961-1971/ [49] Voyager and Three Body Problem, BBC Horizon documentary, an Interview of Michael Minovitch, https://www.youtube.com/watch?v=h66o 5efSd4 [50] Burt Totaro, Wikipedia: http://en.wikipedia.org/wiki/Burt Totaro [51] The Shoshichi Kobayashi Memorial Website: http://www.shoshichikobayashi.com/ (English) http://jp.shoshichikobayashi.com/ (Japanese) Academic Genealogy of Shoshichi Kobayashi 37

[52] Remembrance speeches at the Memorial Service of Shoshichi Kobayashi, September 8, 2012. http://www.shoshichikobayashi.com/remembrances/ [53] Condolence Message/Poem/Memoir: http://www.shoshichikobayashi.com/condolence-message-poem-memoir/ (English) [54] Shoshichi Kobayashi Memorial Symposium: http://www.shoshichikobayashi.com/more-than-130-mathematicians-participate- in-the-shoshichi-kobayashi-memorial-symposium/ [55] The Memorial Symposium’s Programs and Abstracts: http://www.math.sci.osaka-u.ac.jp/ enoki/symp/Kobayashi2013/ [56] Shoshichi Kobayashi Memorial Reception: http://www.shoshichikobayashi.com/the-memorial-reception-attended- by-more-than-one-hundred/ [57] Gary Jensen, “Shoshichi Kobayashi as Advisor,” at the Memorial Reception held at the University of Tokyo, May 25, 2013. http://www.shoshichikobayashi.com/2013/07/25/shoshichi-kobayashi- as-adviser-prof-emeritus-gary-jensen/ [58] Shing-Tung Yau, “In Memory of Prof. Kobayashi,” at the Memorial Reception held at the University of Tokyo, May 25, 2013. http://www.shoshichikobayashi.com/2013/07/25/in-memory-of- prof-kobayashi-prof-shing-tung-yau/ [59] Shing-Tung Yau, “Poem: The wise scholar has now passed away, . . . ” May 25, 2013. http://www.shoshichikobayashi.com/condolence-message-poem-memoir/ [60] “In Memory of Shoshichi Kobayashi,” ICCM Notices, December 2013, pp. 138-158. [61] Gary R. Jensen, Editor “Remembering Shoshichi Kobayashi,” Notices of AMS (scheduled to appear in the December 2014 issue). Post Script With the assistance of my academic sibling Prof. Vince Poor of Princeton Univer- sity, I have found that we are academic descendants of Carl Friedrich Gauss (1777– 1855): Our Ph.D. thesis advisor at Princeton was Prof. John Bowman Thomas: John Bowman Thomas (1955, Stanford University) Willis W. Harman (1948, Stanford University) Karl Ralph Spangenberg (1937, Ohio State University) William Littell Everitt (1933, Ohio State University) Frederick Columbus Blake (1906, Columbia University) Ernest Fox Nichols (1897, Cornell University) Edward Leamington Nichols (1879, Universität Göttingen) Johann Benedict Listing (1834, Universität Göttingen) Carl Friedrich Gauss (1799, Universität Helmstedt) Dr. Michael Gerver of Technion confirmed through the archival office of the Uni- versität Göttingen that Johann Listing was the thesis advisor of the American physicist Edward Nichols. Both the Wikipedia and the Mathematics Genealogy project incorrectly record Hermann von Helmholtz (1821–1894) as Nichols’ advisor. Helmholtz was a professor of Universität Berlin, not Göttingen. Nichols worked 38 H. Kobayashi with Helmholtz at Berlin before he moved to Göttingen, where he submitted his doctoral dissertation with Listing as the advisor.

Hisashi Kobayashi The Sherman Fairchild Professor Emeritus Department of Electrical Engineering Princeton University Princeton NJ 08544, USA